Vinnfred, thank you for your concise and clear answer. I had not seen this explanation anywhere else.
Why would you use a definite article for one and an indefinite article for the other?
in the context of a performance, there is usually one director and multiple actors/actresses.
You usually use або in affirmative sentences (Вона режисер або актриса) in the meaning "I am not sure who she is, but she must be either one or the other (and not both)". You know, like "It's either me or him!" (або я, або він). But it's not typically used in questions